Probabilistic Constellation Shaping of Multi-Dimensional Symbols for Improved Tolerance to Nonlinear Impairments

ABSTRACT

An optical transmitter device includes a digital signal processor (DSP) having digital hardware. The DSP is operative to generate shaped bits from a first set of information bits, and to apply a systematic forward error correction (FEC) scheme to encode the shaped bits and a second set of information bits, where the first set of information bits and the second set of information bits are disjoint sets. Unshaped bits and the shaped bits are mapped to selected symbols or are used to select symbols from one or more constellations. The selected symbols are mapped to physical dimensions. Each unshaped bit is either one of the second set of information bits or one of multiple parity bits resulting from the FEC encoding. In this manner, a target spectral efficiency is achieved.

CROSS-REFERENCE

This application claims the benefit of U.S. patent application Ser. No.62/744,943 filed Oct. 12, 2018, the entire contents of which areincorporated herein by reference.

TECHNICAL FIELD

This document relates to the technical field of probabilisticconstellation shaping.

BACKGROUND

Probabilistic constellation shaping (PCS) is a technique that allowscontrol over the visitation probabilities of different constellationsymbols, yielding unequal visitation probabilities. PCS allows thenumber of bits per symbol (sometimes referred to as the “spectralefficiency”) to be varied in a (nearly) continuous manner withoutrequiring support for multiple discrete constellations to beimplemented. When the probability distribution of the visitationprobabilities is Gaussian, PCS improves the linear noise tolerancerelative to standard modulation having equal visitation probabilitiesfor all symbols.

Various PCS implementations are known in the art, including, forexample, the implementations described in U.S. Pat. No. 9,698,939, andin J. Cho et al., “Low-complexity shaping for enhanced nonlinearitytolerance”, Proceedings of ECOC 2016, pp. 467-469.

In some implementations, the constellation that is shaped is ahigh-cardinality quadrature amplitude modulation (QAM) format such as 64QAM or 256 QAM, and the probability of selecting a given symboldecreases with that symbol's energy. More specifically, the probabilityof selecting the i-th symbol S_(i) in the constellation alphabet isproportional to exp(−μ|S_(i)|²) for a constant μ that depends on thetarget bits per symbol.

QAM formats significantly increase nonlinear interference onlow-dispersion metro or submarine applications.

SUMMARY

Some modulation formats have constant symbol energy, in that the symbolmodulus |S_(i)|² is the same for all symbols, which improves nonlinearperformance by minimizing variations in the amplitude of the opticalfield. Examples of such modulation formats include binary phase shiftkeying (BPSK), quadrature phase shift keying (QPSK), andpolarization-balanced 8-dimensional formats (described below).Polarization-balanced 8-dimensional formats have the additional propertythat the polarization vector in the first time slot is orthogonal tothat of the second time slot. Polarization balancing leads to a partialcancellation of cross-polarization modulation, with the most benefitobserved on dispersion-compensated submarine system. However, with aconstant symbol energy, there is only a single data rate.

This document describes novel constellations for bit-to-symbol mappingand symbol-to-bit demapping that make use of shaped bits and unshapedbits to achieve a target spectral efficiency (defined as a number ofinformation bits per time slot) while preserving constellation qualitiesthat are beneficial for tolerance to nonlinear effects. Such qualitiesinclude power-balancing (that is, constant symbol energy) andpolarization-balancing.

BRIEF DESCRIPTION OF THE DRAWINGS AND APPENDIX

FIG. 1 illustrates an example coherent optical communications systemthat employs polarization-division multiplexing (PDM);

FIG. 2 illustrates functionality of an example transmitter digitalsignal processor;

FIG. 3 illustrates functionality of an example receiver digital signalprocessor;

FIG. 4 and FIG. 5 illustrate two example labeling schemes for aquadrature phase shift keying (QPSK) constellation;

FIG. 6 is a flowchart illustration of an example method fortransmission;

FIG. 7 is a flowchart illustration of an example method for handling twoQPSK symbols that have been mapped to four physical dimensions;

FIG. 8 illustrates a super-constellation formed as the union of an 8PSKconstellation having a larger amplitude and a QPSK constellation havinga smaller amplitude;

FIG. 9 illustrates a super-constellation formed as the union of a QPSKconstellation having a larger amplitude and a QPSK constellation havinga smaller amplitude;

FIG. 10 is a flowchart illustration of an example method for opticaltransmission using the super-constellation formed as the union of anouter constellation of constant energy and an inner constellation ofconstant energy;

FIG. 11 is a flowchart illustration of another example method fortransmission;

FIG. 12 is a flowchart illustration of an example method for handlingfour QPSK symbols that have been mapped to eight physical dimensions;

FIG. 13 is an illustration of pseudo-energies of an artificialconstellation;

FIG. 14 is a flowchart illustration of a method for using a structure togenerate shaped bits;

FIG. 15 is an illustration of an example tree structure; and

Appendix provides an example partition of QPSK_(AUG) ³ into fourdisjoint subsets satisfying a Euclidean distance constraint.

DETAILED DESCRIPTION

Throughout the application, ordinal numbers (e.g., first, second, third,etc.) may be used as an adjective for an element (that is, any noun inthe application). The use of ordinal numbers is not to imply or createany particular ordering of the element unless expressly disclosed, suchas by use of the terms “before”, “after”, and other such terminology.The use of ordinal numbers is not to limit any element to being only asingle element unless expressly disclosed, such as by use of the term“single” and other such terminology. Rather, the use of ordinal numbersis to distinguish between the elements. By way of an example, a firstelement is distinct from a second element, and the first element mayencompass more than one element and succeed (or precede) the secondelement in an ordering of elements.

FIG. 1 illustrates an example coherent optical communications system 10that uses phase modulation (and optionally amplitude modulation) of anoptical carrier signal 12 to convey information. The system employspolarization-division multiplexing (PDM).

A transmitter device 14 (“transmitter 14”) in the example coherentcommunications system 10 comprises a laser 16 to produce the opticalcarrier signal 12, and a polarization beam splitter 18 to split theoptical carrier signal 12 into two orthogonally-polarized components13,15.

A digital signal processor (DSP) 20 in the transmitter 14 generates twopairs of in-phase (I) digital drive signals 22 and quadrature (Q)digital drive signals 24 from information bits 26. Digital-to-analogconverters 28 convert the I/Q digital drive signals 22, 24 intorespective I/Q analog drive signals 32, 34. Each pair of the I/Q analogdrive signals 32, 34 drives a respective electrical-to-optical modulator36 (for example, a Mach-Zehnder modulator) to modulate a respective oneof the orthogonally-polarized components 13,15 of the optical carriersignal 12, thereby generating a respective modulated polarized opticalsignal 17,19. The transmitter 14 comprises a polarization beam combiner38 that combines the two modulated polarized optical signals 17,19,thereby yielding a modulated optical carrier signal 40 for transmissionon an optical link 50. For simplicity, additional components of thetransmitter 14 such as amplifiers are not illustrated or discussed inthis document.

The optical link 50 may comprise one or more of optical fibers, opticalamplifiers, repeaters, wavelength selection switch (WSS) nodes, opticaladd-drop multiplexers (OADMs), reconfigurable OADMs (ROADMs), andoptical power taps for link monitoring and diagnostics.

The information bits 26 to be conveyed by the modulated carrier do notnecessarily have any particular underlying spectral structure. Theinformation bits 26 may be completely arbitrary. The information bits 26may be encrypted data bits resulting from the application ofconventional encryption techniques (with or without built-inauthentication). The information bits 26 may have been scrambled usingany suitable scrambling technique.

The transmitter DSP 20 is digital hardware 30 supported bysoftware/firmware stored in a memory (not shown). An example of thefunctionality implemented by the transmitter DSP 20 is illustrated inFIG. 2. In this example, the functionality includes probabilisticconstellation shaping (PCS) 42, forward error correction (FEC) encoding44 according to a systematic FEC scheme, mapping 46 of FEC-encoded bitsto symbols, processing 48 of the symbols, and extraction of the I/Qcomponents of the processed symbols as the I/Q digital drive signals foreach polarization 22, 24.

A portion of the information bits (denoted a “first set of informationbits”) are shaped. Various PCS implementations are known in the art,including, for example, the implementations described in U.S. Pat. No.9,698,939, and in J. Cho et al., “Experimental demonstration ofphysical-layer security in a fiber-optic link by informationscrambling”, ECOC 2016. Development of new PCS implementations isongoing.

Another portion of the information bits (denoted a “second set ofinformation bits”) remain unshaped. The first set of information bitsand the second set of information are disjoint sets, which means noinformation bit belongs to both the first set and the second set.

The FEC encoding 44 is applied to the shaped bits and to the second setof information bits. Systematic FEC encoding 44 preserves the shaping ofthe shaped bits. The output of the systematic FEC encoding 44 isFEC-encoded bits, which consist of the shaped bits, the second set ofinformation bits, and multiple parity bits. The multiple parity bitshave a probability of being equal to 1 substantially equal to 0.5.

Returning now to FIG. 1, a receiver device 54 (“receiver 54”) in theexample coherent optical communications system 10 comprises a laser 56to generate a local version 58 of the optical carrier signal 12. Thereceiver 54 is operative to receive a received modulated optical carriersignal 60. The received modulated optical carrier signal 60 is adistorted version of the modulated optical carrier signal 40 that wastransmitted by the transmitter 14. The receiver comprises a polarizationbeam splitter 62 to split the received modulated optical carrier signal60 into two orthogonally-polarized components. An optical hybrid 64mixes the local version 58 of the optical carrier signal with theorthogonally-polarized components of the received modulated carriersignal 60, and photodetectors 66 are used to yield two pairs of I/Qanalog signals 72, 74. Analog-to-digital converters 76 sample the pairsof I/Q analog signals 72, 74 to yield respective pairs of I/Q digitalsignals 82, 84. A DSP 78 in the receiver 54 processes the two pairs ofI/Q digital signals 82, 84 to yield recovered bits 86. For simplicity,additional components of the receiver 54 such as amplifiers are notillustrated or discussed in this document.

In an ideal coherent communications system 10, the recovered bits 86 areidentical to the information bits 26. In practice, however, impairmentsin the transmitter 14, the link 50, and the receiver 54 cause errors inthe recovered bits 86.

The receiver DSP 78 is digital hardware 80 supported bysoftware/firmware stored in a memory (not shown). An example of thefunctionality implemented by the receiver DSP 78 is illustrated in FIG.3. In this example, the functionality includes polarization recovery 90which identifies the X- and Y-polarizations of the modulated opticalcarrier signal, down-sampling 91 to extract the received optical symbolsfrom the I/Q digital signals 82, 84, carrier recovery 92 which correctsvariation in the phase of the received optical symbols, de-mapping 94 ofthe processed symbols to FEC-encoded bits, FEC decoding anderror-correction 96 of the FEC-encoded bits, and inverse shaping 98 of aportion of the error-corrected FEC-decoded bits to yield, together withthe portion of the error-corrected FEC-decoded bits that do not requireinverse shaping, the recovered bits 86.

This document describes novel constellations for bit-to-symbol mappingand symbol-to-bit demapping that make use of shaped bits and unshapedbits to achieve a target spectral efficiency (defined as a number ofinformation bits per time slot) while preserving constellation qualitiesthat are beneficial for tolerance to nonlinear effects. Such qualitiesinclude power-balancing and polarization-balancing, as demonstrated inA. Shiner et al., “Demonstration of an 8-dimensional modulation formatwith reduced inter-channel nonlinearities in a polarization multiplexedcoherent system”, Opt. Expr. vol. 22, issue 17, pp.20366-20374 (2014)and M. Reimer et al., “Optimized 4 and 8 dimensional modulation formatsfor variable capacity in optical networks”, In Optical FiberCommunication Conference 2016, OSA Technical Digest (online), M3A.4.Polarization balancing is also described in U.S. Pat. No. 9,143,238,entitled “Optical modulation schemes having reduced nonlinear opticaltransmission impairments”.These constellations and bit-to-symbol mappingand symbol-to-bit demapping techniques may be suitable for applicationshaving spectral efficiencies nominally between 1 and 6 bits per timeslot appropriate for low net dispersion optical systems for regional orsubmarine transmission distances.

This document describes how to adapt tree-encoder PCS for shaping bitsto be used with the novel constellations.

A normalized quadrature phase shift keying (QPSK) constellation has analphabet of four complex-valued symbols, for example

$\left\{ {{S_{1} = {\exp \left( {i\frac{\pi}{4}} \right)}},{S_{2} = {\exp \left( {i\frac{3\pi}{4}} \right)}},{S_{3} = {\exp \left( {i\frac{5\pi}{4}} \right)}},{S_{4} = {\exp \left( {i\frac{7\pi}{4}} \right)}}} \right\} \mspace{14mu} {or}${S₁ = 1, S₂ = −1, S₃ = i, S₄ = −i}.

The complex-valued symbols have the same energy and differ only inphase. Each complex-valued symbol can be labelled using 2 bits. FIG. 4and FIG. 5 illustrate two example labeling schemes for the QPSKconstellation.

FIG. 6 is a flowchart of an example method for transmission. A targetspectral efficiency between 2 and 4 bits per time slot may be achievedwith the method illustrated in FIG. 6. for selecting QPSK symbols. TheDSP 20 generates (102) shaped bits from a first set of information bits.The DSP 20 applies (104) a systematic FEC scheme to encode the shapedbits and a second set of information bits, where the first set ofinformation bits and the second set of information bits are disjointsets.

The DSP 20 maps (106) a first group of two bits to a first symbol of aQPSK constellation according to a labeling scheme. The first groupconsists of a first unshaped bit and a first one of the shaped bits. TheDSP 20 maps (108) a second group of two bits to a second symbol of aQPSK constellation according to the labeling scheme. The second groupconsists of a second unshaped bit and a second one of the shaped bits.Each unshaped bit is either one of the second set of information bits orone of multiple parity bits resulting from the FEC encoding.

The DSP 20 maps (110) the first QPSK symbol and the second QPSK symbolto four physical dimensions. The physical dimensions may be spreadacross any suitable combination of polarization, time, carrier (orsubcarrier) wavelength, fiber propagation mode, or core within amulti-core fiber. For example, the four physical dimensions may beamplitude and phase (or equivalently, in-phase and quadrature) of twoorthogonally-polarized components of an optical carrier signal.

The DSP 20 selects (112) another pair consisting of a new first groupand a new second group and proceeds to handle the bits of those groupsas described above.

FIG. 7 is a flowchart of an example method for handling two QPSK symbolsthat have been mapped to four physical dimensions. The polarization beamsplitter 18 splits (114) an optical carrier into twoorthogonally-polarized components. In a single time slot, one of theelectrical-to-optical modulators 36 modulates (116) theorthogonally-polarized component 13 to convey the first QPSK symbol (ora processed version thereof), and the other of the electrical-to-opticalmodulators 36 modulates (116) the orthogonally-polarized component 15 toconvey the second QPSK symbol (or a processed version thereof). Thepolarization beam combiner 38 combines (405) the two modulated polarizedoptical signals 17,19 for transmission.

Let b₀ denote the shaped bit and b₁ denote the unshaped bit. Let P(0)denote the probability of the shaped bit b₀ being equal to 0, and P(1)denote the probability of the shaped bit b₀ being equal to 1.

In one implementation, the most significant bit in all groups is alwaysone of the shaped bits. Then b₀b₁ is the two-bit group that is mapped toone of the symbols of the QPSK constellation. Suppose the probabilityP(0) is 0.65 and the probability P(1) is 0.35. With the labeling schemeillustrated in FIG. 4 or the labeling scheme illustrated in FIG. 5, theQPSK symbols illustrated as black circles will be visited more oftenthan the QPSK symbols illustrated as white circles, because the shapedbit b₀ is the most significant bit and is more likely to be equal to 0than equal to 1.

In another implementation, the least significant bit in all groups isalways one of the unshaped bits. Then b₁b₀ is the two-bit group that ismapped to one of the symbols of the QPSK constellation. Suppose theprobability P(0) is 0.8 and the probability P(1) is 0.2. With thelabeling scheme illustrated in FIG. 4, the QPSK symbols to the right ofthe Q axis will be visited more often than the QPSK symbols to the leftof the Q axis, because the shaped bit b₀ is the least significant bitand is much more likely to be equal to 0 than equal to 1. With thelabeling scheme illustrated in FIG. 5, the QPSK symbols above the I axiswill be visited more often than the QPSK symbols below the I axis,because the shaped bit b₀ is the least significant bit and is much morelikely to be equal to 0 than equal to 1.

In circumstances where the probability P(0) is 0.5, all symbols of theQPSK constellation are visited with equal probability, and the spectralefficiency is 4 bits per time slot in a dual-polarization system.

In circumstances where the probability P(0) is 0 or 1, two symbols ofthe QPSK constellation are visited with equal probability and the othertwo symbols are never visited, so effectively this is similar to a BPSKconstellation, and the spectral efficiency is 2 bits per time slot in adual-polarization system.

In circumstances where the probability P(0) satisfies 0<P(0)<0.5 or0.5<P(0)<1, the spectral efficiency is between 2 and 4 bits per timeslot in a dual polarization system.

The spectral efficiency ϵ, in units of bits per time slot, may becalculated as follows.

ϵ=2·[1−P(0)log₂ P(0)−P(1)log₂ P(1)]  (1)

FIG. 8 illustrates a super-constellation formed as the union of an 8PSKconstellation whose symbols have an amplitude A and a QPSK constellationwhose symbols have an amplitude B. For convenience, thissuper-constellation is referred to as the “outer 8PSK-inner QPSK”super-constellation. A and B are real positive numbers subject to theconstraints that B<A and A²+B²=1, as discussed in U.S. Pat. No.9,749,058, entitled “Nonlinear tolerant optical modulation formats athigh spectral efficiency”. In practice, the sum of squares may differslightly from 1. For example, the sum A²+B² may have a value in therange between 0.95 and 1.05, inclusive. A normalized 8 phase shiftkeying (8PSK) constellation has an alphabet of eight complex-valuedsymbols, for example

$\left\{ {{U_{1} = {\exp \left( {i\frac{\pi}{8}} \right)}},{U_{2} = {\exp \left( {i\frac{3\pi}{8}} \right)}},{U_{3} = {\exp \left( {i\frac{5\pi}{8}} \right)}},{U_{4} = {\exp \left( {i\frac{7\pi}{8}} \right)}},{U_{5} = {\exp \left( {i\frac{9\pi}{8}} \right)}},{U_{6} = {\exp \left( {i\frac{11\pi}{8}} \right)}},{U_{7} = {\exp \left( {i\frac{13\pi}{8}} \right)}},{U_{8} = {\exp \left( {i\frac{15\pi}{8}} \right)}}} \right\}.$

In practice, the phases of the complex-valued symbols may be optimizedfor improved tolerance to additive noise. The complex-valued symbolshave the same power and differ only in phase. Each complex-valued symbolcan be labelled using 3 bits. Example labeling schemes for the 8PSKconstellation and for the QPSK constellation are illustrated in FIG. 8.

FIG. 9 illustrates a super-constellation formed as the union of an outerQPSK constellation whose symbols have an amplitude A and an inner QPSKconstellation whose symbols have an amplitude B. For convenience, thissuper-constellation is referred to as the “outer QPSK-inner QPSK”super-constellation. A and B are real positive numbers subject to theconstraints that B<A and A²+B²=1. In practice, the sum of squares maydiffer slightly from 1. For example, the sum A²+B² may have a value inthe range between 0.95 and 1.05, inclusive. Example labeling schemes forboth QPSK constellations are illustrated in FIG. 9.

FIG. 10 is a flowchart illustration of an example method for opticaltransmission using a super-constellation formed as the union of an outerconstellation of constant energy and an inner constellation of constantenergy.

The DSP 20 generates (202) shaped bits from a first set of informationbits. The DSP 20 applies (204) a systematic FEC scheme to encode theshaped bits and a second set of information bits, where the first set ofinformation bits and the second set of information bits are disjointsets.

The DSP 20 then considers a group consisting of K unshaped bits and oneof the shaped bits. Each unshaped bit is either one of the second set ofinformation bits or one of multiple parity bits resulting from the FECencoding. For the “outer 8PSK-inner QPSK” super-constellation, K equals5. For the “outer QPSK-inner QPSK” super-constellation, K equals 4.

The DSP 20 selects (206) one of two constellations based on a value ofthe shaped bit in the group. All symbols of one of the twoconstellations have an amplitude A, and all symbols of the other of thetwo constellations have an amplitude B, where A and B are real positivenumbers subject to the constraints that B<A and A²+B²=1. In practice,the sum of squares may differ slightly from 1. For example, the sumA²+B² may have a value in the range between 0.95 and 1.05, inclusive.The total number of symbols in the “outer 8PSK-inner QPSK”super-constellation is 2³×2²=32. The total number of symbols in the“outer QPSK-inner QPSK” super-constellation is 2²×2²=16.

The DSP 20 uses the K unshaped bits to select (208) a first symbol froma first of the two constellations and a second symbol from a second ofthe two constellations.

For the “outer 8PSK-inner QPSK” super-constellation, the DSP 20 mapsthree of the five unshaped bits in the group to an 8PSK symbol havingthe amplitude A and maps a remaining two of the five unshaped bits inthe group to a QPSK symbol having the amplitude B.

For the “outer QPSK-inner QPSK” super-constellation, the DSP 20 maps twoof the four unshaped bits in the group to a QPSK symbol having theamplitude A and maps a remaining two of the four unshaped bits in thegroup to a QPSK symbol having the amplitude B.

Depending on the value of the shaped bit in the group, either the DSP 20maps (210) the first symbol to a first polarization and the secondsymbol to a second polarization that is orthogonal to the firstpolarization, or the DSP 20 maps (212) the second symbol to the firstpolarization and the first symbol to the second polarization. Morespecifically, the DSP 20 maps one of the symbols to two physicaldimensions of an optical carrier component that is polarized in thefirst polarization, and maps the other of the symbols to two physicaldimensions of an optical carrier component that is polarized in thesecond polarization. For example, the two physical dimensions may beamplitude and phase (or equivalently, in-phase and quadrature) of apolarized component of an optical carrier signal.

For the “outer 8PSK-inner QPSK” super-constellation, depending on thevalue of the shaped bit in the group, the DSP maps the 8PSK symbolhaving the amplitude A to one polarization and the QPSK symbol havingthe amplitude B to the orthogonal polarization.

For the “outer QPSK-inner QPSK” super-constellation, depending on thevalue of the shaped bit in the group, the DSP maps the QPSK symbolhaving the amplitude A to one polarization and the QPSK symbol havingthe amplitude B to the orthogonal polarization.

The DSP 20 selects (214) another group consisting of another K unshapedbits and another one of the shaped bits, and proceeds to handle the bitsof that group as described above.

Let b₀ denote the shaped bit and b₁ denote the unshaped bit. Let P(0)denote the probability of the shaped bit b₀ being equal to 0, and P(1)denote the probability of the shaped bit b₀ being equal to 1.

For the “outer 8PSK-inner QPSK” super-constellation, the spectralefficiency c may be calculated as given in Equation 2.

ϵ=log₂(32)−P(0)log₂ P(0)−P(1) log₂ P(1)  (2)

In circumstances where the probability P(0)=P(1)=0.5, the spectralefficiency is 6 bits per time slot in a dual-polarization system. Incircumstances where the probability P(0) is 0 or 1, the spectralefficiency is 5 bits per time slot in a dual-polarization system. Incircumstances where the probability P(0) satisfies 0<P(0)<0.5 or0.5<P(0)<1, the spectral efficiency is between 5 and 6 bits per timeslot in a dual polarization system.

Accordingly, the method illustrated in FIG. 10, when used with the“outer 8PSK-inner QPSK” super-constellation illustrated in FIG. 8, mayachieve a target spectral efficiency between 5 and 6 bits per time slotin a dual polarization system.

For the “outer QPSK-inner QPSK” super-constellation, the spectralefficiency ϵ may be calculated as given in Equation 3.

ϵ=log₂(16)−P(0)log₂ P(0)−P(1)log₂ P(1)  (3)

In circumstances where the probability P(0)=P(1)=0.5, the spectralefficiency is 5 bits per time slot in a dual-polarization system. Incircumstances where the probability P(0) is 0 or 1, the spectralefficiency is 4 bits per time slot in a dual-polarization system. Incircumstances where the probability P(0) satisfies 0<P(0)<0.5 or0.5<P(0)<1, the spectral efficiency is between 4 and 5 bits per timeslot in a dual polarization system.

Accordingly, the method illustrated in FIG. 10, when used with the“outer QPSK-inner QPSK” super-constellation illustrated in FIG. 9, mayachieve a target spectral efficiency between 4 and 5 bits per time slotin a dual polarization system.

Similarly, the method illustrated in FIG. 10, when used with an “outer8PSK-inner 8PSK” super-constellation (not illustrated), may achieve atarget spectral efficiency between 6 and 7 bits per time slot in a dualpolarization system. This combination may result in a significantpenalty in terms of tolerance to additive noise, which limitspracticality.

The two QPSK symbols selected using the method illustrated in FIG. 10are mapped to four physical dimensions and may be handled using theexample method illustrated in FIG. 7 and discussed above.

FIG. 11 is a flowchart of another example method for transmission. TheDSP 20 generates (302) shaped bits from a first set of information bits.The DSP 20 applies (304) a systematic FEC scheme to encode the shapedbits and a second set of information bits, where the first set ofinformation bits and the second set of information bits are disjointsets.

The DSP then considers a group consisting of K unshaped bits and L ofthe shaped bits. Each unshaped bit is either one of the second set ofinformation bits or one of multiple parity bits resulting from the FECencoding.

The DSP 20 selects (306) one of 2^(L) collections based on a value ofthe L shaped bits in the group. Examples of collections are providedbelow.

The DSP 20 uses (308) the K unshaped bits to select an ordered N-tupleof QPSK symbols from the selected collection. For example, where Nequals 3, the DSP 20 uses the K unshaped bits to select an orderedtriplet of QPSK symbols from the selected collection. In anotherexample, where N equals 4, the DSP 20 uses the K unshaped bits to selectan ordered quadruplet of QPSK symbols from the selected collection.

The DSP 20 then maps (310) four QPSK symbols to eight physicaldimensions. The physical dimensions may be spread across any suitablecombination of polarization, time, carrier (or subcarrier) wavelength,fiber propagation mode, or core within a multi-core fiber. For example,the eight physical dimensions may be amplitude and phase (orequivalently, in-phase and quadrature) of two orthogonally-polarizedcomponents of an optical carrier signal of two different time slots. Inanother example, the eight physical dimensions may be amplitude andphase (or equivalently, in-phase and quadrature) of twoorthogonally-polarized components of two different subcarriers of anoptical signal. In examples where N equals 4, the four QPSK symbols thatare mapped to eight physical dimensions are the four QPSK symbols of theselected ordered quadruplet of QPSK symbols. In examples where N equals3, the four QPSK symbols that are mapped to eight physical dimensionsinclude the three QPSK symbols of the selected ordered triplet of QPSKsymbols. A fourth QPSK symbol is determined as follows. Let the QPSKsymbols in the selected ordered triplet be denoted X₁, Y₁, and X₂,respectively. The fourth QPSK symbol denoted Y₂ satisfies a conditionX₁Y₁*+X₂Y₂*=0, where asterisk (*) denotes complex conjugate. Where thefour QPSK symbols are mapped to orthogonally-polarized components of anoptical carrier signal across two time slots, the conditionX₁Y₁*+X₂Y₂*=0 is referred to as “polarization balanced” and the fourQPSK symbols are referred to as “polarization-balanced QPSK symbols”.

The DSP 20 selects (312) another group consisting of another K unshapedbits and another L of the shaped bits, and proceeds to handle the bitsof that group as described above.

FIG. 12 is a flowchart of an example method for handling four QPSKsymbols that have been mapped to eight physical dimensions. Thepolarization beam splitter 18 splits (402) an optical carrier into twoorthogonally-polarized components. In a first time slot, one of theelectrical-to-optical modulators 36 modulates (404) theorthogonally-polarized component 13 to convey the first QPSK symbol (ora processed version thereof), and the other of the electrical-to-opticalmodulators 36 modulates (404) the orthogonally-polarized component 15 toconvey the second QPSK symbol (or a processed version thereof). Thepolarization beam combiner 38 combines (406) the two modulated polarizedoptical signals 17,19 for transmission. In a second time slot, one ofthe electrical-to-optical modulators 36 modulates (408) theorthogonally-polarized component 13 to convey the third QPSK symbol (ora processed version thereof), and the other of the electrical-to-opticalmodulators 36 modulates (408) the orthogonally-polarized component 15 toconvey the fourth QPSK symbol (or a processed version thereof). Thepolarization beam combiner 38 combines (410) the two modulated polarizedoptical signals 17,19 for transmission. The next four QPSK symbols areidentified (412) and the method resumes from a new “first” time slot.When modulated on to the optical carrier, symbols are effectively“spread out” in time due to the impulse response of the transmitter.That is, the digital symbols are contained within a single time slot,but this digital waveform is up-sampled to an analog waveform with acertain transmitter pulse shape. After modulation, the symbols overlapin time, and the instruction to the modulator at any instant is a sum ofa number of up-sampled symbols. The wording “or a processed versionthereof” in the foregoing description is intended to cover thiscondition.

Euclidean Distance Shaped (EDS)-256-PB-QPSK-Spectral Efficiency Between3.5 and 4 Bits per Time Slot

Consider a fourth-order Cartesian product of a QPSK constellation:QPS⁽⁴⁾=QPSK⊗QPSK⊗QPSK⊗QPSK. The elements of QPSK⁽⁴⁾ are all 256 possibleordered quadruplets of QPSK symbols. A labeling scheme uses eight bitsto label all possible ordered quadruplets of QPSK symbols. QPSK⁽⁴⁾ canbe partitioned into two collections. One collection, denoted “the evenparity collection”, consists of all 128 ordered quadruplets of QPSKsymbols whose 8-bit label according to the labeling scheme has evenparity. The other collection, denoted “the odd parity collection”,consists of all 128 ordered quadruplets of QPSK symbols whose 8-bitlabel according to the labeling scheme has odd parity.

For this example, the method illustrated in FIG. 11 is used where Kequals 7, L equals 1 (hence two collections), and N equals 4 (henceordered quadruplets of QPSK symbols).

Each group consists of 7 unshaped bits and one shaped bit. The value ofthe one shaped bit determines whether the even parity collection isselected or the odd parity collection is selected. The 7 unshaped bitsare used to select from among the 128 ordered quadruplets of QPSKsymbols in the selected collection. The four QPSK symbols in theselected ordered quadruplet are mapped to eight physical dimensions.

For this example, the spectral efficiency c over two time slots may becalculated as given in Equation 4.

ϵ=7−P(0) log₂ P(0)−P(1) log₂ P(1)  (4)

Accordingly, the method illustrated in FIG. 11, when used with thisexample, may achieve a target spectral efficiency between 3.5 and 4 bitsper time slot in a dual polarization system.

EDS-128-PB-QPSK—Spectral Efficiency Between 3 and 3.5 Bits per Time Slot

Denote the symbols of a conventional QPSK constellation as S₁, S₂, S₃,and S₄.

Define a right-rotated QPSK symbol as a conventional QPSK symbolmultiplied by

${\exp \left( {i\frac{\pi}{8}} \right)}.$

Stated differently, the right-rotated QPSK constellation, QPSK_(RIGHT),has the symbols

$\left\{ {{{\exp \left( {i\frac{\pi}{8}} \right)}S_{1}},{{\exp \left( {i\frac{\pi}{8}} \right)}S_{2}},{\exp \left( {i\frac{\pi}{8}} \right)S_{3}},{\exp \left( {i\frac{\pi}{8}} \right)S_{4}}} \right\}.$

Define a left-rotated QPSK symbol as a conventional QPSK symbolmultiplied by

${\exp \left( {{- i}\frac{\pi}{8}} \right)}.$

Stated differently, the left-rotated QPSK constellation, QPSK_(LEFT),has the symbols

$\left\{ {{{\exp \left( {{- i}\frac{\pi}{8}} \right)}S_{1}},{{\exp \left( {{- i}\frac{\pi}{8}} \right)}S_{2}},{{\exp \left( {{- i}\frac{\pi}{8}} \right)}S_{3}},{{\exp \left( {{- i}\frac{\pi}{8}} \right)}S_{4}}} \right\}.$

One collection is a third-order Cartesian product of a right-rotatedQPSK constellation: QPSK⁽³⁾_(RIGHT)=QPSK_(RIGHT)⊗QPSK_(RIGHT)⊗QPSK_(RIGHT). The elements of QPSK⁽³⁾_(RIGHT) are all 64 possible ordered triplets of right-rotated QPSKsymbols.

The other collection is a third-order Cartesian product of aleft-rotated QPSK constellation: QPSK⁽³⁾_(LEFT)=QPSK_(LEFT)⊗QPSK_(LEFT)⊗OQPSK_(LEFT). The elements ofQPSK_(LEFT) are all 64 possible ordered triplets of left-rotated QPSKsymbols.

For this example, the method illustrated in FIG. 11 is used where Kequals 6, L equals 1 (hence two collections), and N equals 3 (henceordered triplets of rotated QPSK symbols).

Each group consists of 6 unshaped bits and one shaped bit. The value ofthe one shaped bit determines whether the right-rotated collection isselected or the left-rotated collection is selected. The 6 unshaped bitsare used to select from among the 64 ordered triplets of rotated QPSKsymbols in the selected collection.

The four QPSK symbols that are mapped to eight physical dimensionsinclude the three rotated QPSK symbols of the selected ordered triplet.A fourth QPSK symbol is determined as follows. Let the rotated QPSKsymbols in the selected ordered triplet be denoted X₁, Y₁, and X₂,respectively. The fourth QPSK symbol denoted Y₂ satisfies a conditionX₁Y₁*+X₂Y₂*=0, where asterisk (*) denotes complex conjugate.

The three rotated QPSK symbols in the selected ordered triplet aremapped to eight physical dimensions.

For this example, the spectral efficiency c over two time slots may becalculated as given in Equation 5.

ϵ=6−P(0)log₂ P(0)−P(1)log₂ P(1)  (5)

Accordingly, the method illustrated in FIG. 11, when used with thisexample, may achieve a target spectral efficiency between 3 and 3.5 bitsper time slot in a dual polarization system.

EDS-64-PB-QPSK—Spectral Efficiency Between 2 and 3 Bits per Time Slot

Consider a third-order Cartesian product of a QPSK constellation:QPS⁽³⁾=QPSK⊗QPSK⊗QPSK⊗QPSK. The elements of QPS⁽³⁾ are all 64 possibleordered triplets of QPSK symbols.

Consider an augmented version of QPS⁽³⁾, denoted QPS⁽³⁾ _(AUG), thatincludes 64 quadruplets of QPSK symbols. In each quadruplet, the firstthree QPSK symbols are identical to the three QPSK symbols that comprisea corresponding one of the 64 possible ordered triplets. Let the threeQPSK symbols be denoted X₁, Y₁, and X₂, respectively. Determine a fourthQPSK symbol denoted Y2 that satisfies a condition X₁Y₁*+X₂Y₂*=0, whereasterisk (*) denotes complex conjugate. Thus for each ordered triplet inQPSK(³) there is a corresponding ordered quadruplet in QPS⁽³⁾ _(AUG).

QPS⁽³⁾ _(AUG) can be partitioned into four disjoint subsets (“fourcollections”), each consisting of a respective 16 of the orderedquadruplets, such that the Euclidean distance between any pair ofordered quadruplets in a given collection is greater than or equal to2√{square root over (2)}. One example partition of QPS⁽³⁾ _(AUG) intofour disjoint subsets satisfying this Euclidean distance constraint isprovided in the Appendix.

As mentioned above, the DSP 20 is configured to generate shaped bitsfrom a first set of information bits. Any PCS implementation may be usedto shape the first set of information bits, thereby yielding the shapedbits.

FIG. 13 is an illustration of an artificial constellation C consistingof two points. FIG. 14 is a flowchart illustration of a method for usinga structure to generate shaped bits. Each point, whether real-valued orcomplex-valued, can be labelled using a single bit b₀. Assign (602)different pseudo-energies to the two points of the artificialconstellation C. For example, assign the pseudo-energy e₁=1 to the pointwith the label b₀=1, and assign the pseudo-energy e₂=3 to the point withthe label b₀=0. Program (604) the structure to output the requirednumber of shaped bits while minimizing the average pseudo-energy. Thestructure programmed in this manner will favor b₀=1 over b₀=0. Stateddifferently, assigning a higher pseudo-energy to the label b₀=0 and alower pseudo-energy to the label b₀=1 ensures that the shaped bit willhave a higher probability P(1) of being equal to 1 and a lowerprobability P(0) of being equal to 0. Changing the pseudo-energiesassigned to the two points will change the probabilities P(1) and P(0).The artificial constellation C is artificial, and the energies assignedto its points are pseudo-energies used only in the programming of thestructure but not mapped to physical dimensions. The artificialconstellation is a construct used to program the structure to achieveparticular probabilities P(0) and P(1) of the shaped bits. In thisexample, the artificial constellation C has two points encoding 1 bit,however the number of artificial constellation points may be extended to2^(N), where N>1, in a similar manner.

In one example, the structure implements a distribution matcher, whichis a DSP algorithm for transforming a sequence of information bits intoa corresponding sequence of shaped bits using, for example, thealgorithm described in P. Schulte, “Constant composition distributionmapping”, IEEE Trans. Info. Theory, vol. 62, no. 1, pp. 430-434 (2016).When programming the distribution matcher structure, the probabilityP(e₁) of pseudo-energy e₁ is set to the desired probability P(1) of b₀being equal to 1, and a lower probability P(0) of b₀ being equal to 0 isassigned to P(e₂).

In another example, the structure is a tree structure, and programmingthe structure involves programming the look-up tables of the treestructure.

Let C⁽²⁾ denote a 2-dimensional constellation constructed as theCartesian product of the artificial constellation C with itself:C⁽²⁾=C⊗C. Each 2-dimensional point S_(k) ⁽²⁾ of C⁽²⁾ is an ordered pairof complex-valued points of the artificial constellation C.

Let C⁽⁴⁾ denote a 4-dimensional constellation constructed as theCartesian product of C⁽²⁾ with itself: C⁽⁴⁾=C⁽²⁾⊗C⁽²⁾. Each4-dimensional point S_(k) ⁽⁴⁾ of C⁽⁴⁾ is an ordered quadruplet ofcomplex-valued points of the artificial constellation C.

Let C⁽⁸⁾ denote an 8-dimensional constellation constructed as theCartesian product of C⁽⁴⁾ with itself: C⁽⁸⁾=C⁽⁴⁾⊗C⁽⁴⁾. Each8-dimensional point S_(k) ⁽⁸⁾ of C⁽⁸⁾ is an ordered octuplet ofcomplex-valued points of the artificial constellation C.

An example tree structure 800 is illustrated in FIG. 15. The exampletree structure 800 is for an 8-dimensional constellation C⁽⁸⁾constructed from the artificial constellation C illustrated in FIG. 13.The example tree structure 800 has five layers: a 1^(st) layer havingsixteen LUTs 801 through 816, a 2^(nd) layer having eight LUTs 821through 828, a 3^(rd) layer having four LUTs 841, 842, 843, 844, a4^(th) layer having two LUTs 881, 882, and a 5^(th) layer having asingle LUT 900. Due to space constraints, only 1^(st)-layer LUTs 801,802, 815 and 816 are illustrated in FIG. 15.

The example tree structure 800 maps 13 information bits to 16 shapedbits, as described in more detail below.

1^(st) Layer of LUTs

Each of the 1^(st)-layer LUTs 801 through 816 maps a 1-bit parent indexto the 1 shaped bit. An example of the 1^(st)-layer LUTs is given inTable 1:

TABLE 1 1-bit Parent Index 1 2 1-Dim Pseudo-Energy e₁ = 1 e₂ = 3 ShapedBit 1 0

The artificial constellation C has two complex-valued points. Thepseudo-energy of each point is also noted in Table 1. A pseudo-energythat is indexed in the 1^(st)-layer LUTs 801 through 816 is denoted a“1-dimensional pseudo-energy”.

2^(nd) Layer of LUTs

There are 4 possible ordered pairs of the 1-dimensional pseudo-energies.The pseudo-energy associated with a pair of 1-dimensionalpseudo-energies is the sum of the 1-dimensional pseudo-energies. Apseudo-energy that is indexed in the 2^(nd)-layer LUTs 821 through 828is denoted a “2-dimensional pseudo-energy”.

An example of the 2^(nd)-layer LUTs 821 through 828 is given in Table 2:

TABLE 2 2-bit Parent Index 1 2 3 4 2-Dim Pseudo-Energy e′₁ = 2 e′₂ = 4e′₃ = 4 e′₄ = 6 1 1 2 2 → 1-bit left child index 1 2 1 2 → 1-bit rightchild index

Each of the 2^(nd)-layer LUTs 821 through 828 maps a 2-bit parent indexto a 1-bit left child index and to a 1-bit right child index. The 2-bitparent index identifies one of the 4 possible 2-dimensionalpseudo-energies. The 1-bit left child index is the index of the first1-dimensional pseudo-energy in the ordered pair, and the 1-bit rightchild index is the index of the second 1-dimensional pseudo-energy inthe ordered pair. For example, if the 2-bit parent index identifies the2-dimensional energy e′₃=4, the 1-bit left child index will identify the1-dimensional pseudo-energy e₂ and the 1-bit right child index willidentify the 1-dimensional pseudo-energy e_(1. Each) 1-bit child indexis the 1-bit parent index of a respective one of the 1^(st)-layer LUTs801 through 816.

3^(rd) Layer of LUTs

There are 16 possible ordered pairs of the 2-dimensionalpseudo-energies. The pseudo-energy associated with a pair of2-dimensional pseudo-energies is the sum of the 2-dimensionalpseudo-energies. In this example, the 3^(rd)-layer LUTs 841, 842, 843,844 implement 1-bit merging by considering the average of two orderedpairs of the 2-dimensional pseudo-energies. For example, the “averageenergy” resulting from merging the first ordered pair of 2-dimensionalpseudo-energies (e₃; e₁) and the second ordered pair of 2-dimensionalpseudo-energies (e₂; e₂) is 7. There are 8 such averages, which areindexed in the 3^(rd)-layer LUTs 841, 842, 843, 844. A pseudo-energythat is indexed in the 3^(rd)-layer LUTs 841, 842, 843, 844 is denoted a“4-dimensional pseudo-energy”. This is referred to as “1-bit merging”because instead of using four bits to index the pseudo-energies of all16 possible ordered pairs of the 2-dimensional pseudo-energies, onlythree bits are required to index the eight 4-dimensionalpseudo-energies.

An example of the 3^(rd)-layer LUTs 841, 842, 843, 844 is given in Table3:

TABLE 3 3-bit Parent Index 1 2 3 4 . . . 7 8 4-Dim Pseudo-Energy 1 InfoBit e″₁ = 5 e″₂ = 6 e″₃ = 7 e″₄ = 8 . . . e″₇ = 10 e″₈ = 11 0 1 2 3 2 .. . 4 4 → 2-bit left 1 1 1 2 3 . . . 3 4 child index 0 1 1 1 3 . . . 2 3→ 2-bit right 1 2 3 2 2 . . . 4 4 child index

Each of the 3^(rd)-layer LUTs 841, 842, 843, 844 maps a singleinformation bit and a 3-bit parent index to a 2-bit left child index andto a 2-bit right child index. The 3-bit parent index identifies one ofthe eight possible 4-dimensional pseudo-energies. The single informationbit selects between the two ordered pairs of 2-dimensionalpseudo-energies that were averaged to yield the identified 4-dimensionalpseudo-energy. The 2-bit left child index is the index of the first2-dimensional pseudo-energy in the ordered pair selected by the singleinformation bit, and the 2-bit right child index is the index of thesecond 2-dimensional pseudo-energy in the ordered pair selected by thesingle information bit. For example, if the 3-bit parent indexidentifies the 4-dimensional pseudo-energy e″₃=7 and the informationbits into the left and right child look-up-tables are both equal to 0,the 2-bit left child index will identify the 2-dimensional pseudo-energye₃ and the 2-bit right child index will identify the 2-dimensionalpseudo-energy e₁. Each 2-bit child index is the 2-bit parent index of arespective one of the 2^(nd)-layer LUTs 821 through 828.

4^(th) Layer of LUTs

There are 64 possible ordered pairs of the 4-dimensionalpseudo-energies. The pseudo-energy associated with a pair of4-dimensional pseudo-energies is the sum of the 4-dimensionalpseudo-energies. In this example, the 4^(th)-layer LUTs 881, 882implement 5-bit clipping. The 5-bit clipping is the act of discarding 32of the ordered pairs of the 4-dimensional pseudo-energies. The orderedpairs that are discarded are those having the highest pseudo-energyassociated therewith. The ordered pairs are discarded in the sense thatthe 4^(th)-layer LUTs 881, 882 do not index the pseudo-energiesassociated with those ordered pairs. This is referred to as “5-bitclipping” because 2⁵ ordered pairs remain after the clipping operation.

In this example, the 4^(th)-layer LUTs 881, 882 implement 2-bit mergingby considering the average of four ordered pairs of the 4-dimensionalpseudo-energies. For example, the “average pseudo-energy” resulting frommerging the four ordered pairs of 4-dimensional pseudo-energies {(e″₁;e″₁), (e″₁; e″₂), (e″₂; (e″₁; e″₃)} is 11. In another example, the“average pseudo-energy” resulting from merging the four ordered pairs of4-dimensional pseudo-energies {(e″₃; e″₁), (e″₂; e″₂), (e″₂; e″.₃),(e″₃; e″₂)} is 12.5. In another example, the “average pseudo-energy”resulting from merging the four ordered pairs of 4-dimensionalpseudo-energies {(e″₁; e″₄), (e″₄; e″₁), (e″₁; e″₅), (e″₅; e″₁)} is 13.There are eight such averages, which are indexed in the 4^(th)-layerLUTs 881, 882. A pseudo-energy that is indexed in the 4^(th)-layer LUTs881, 882 is denoted an “8-dimensional energy”. This is referred to as“2-bit merging”, because instead of using 5 bits to index thenot-discarded 32 ordered pairs of the 4-dimensional pseudo-energies,only 3 bits are required to index the eight 8-dimensionalpseudo-energies.

An example of the 4^(th)-layer LUTs 881, 882 is given in Table 4:

TABLE 4 3-bit Parent Index 1 2 3 4 . . . 8 8-Dim Pseudo-Energy 2 InfoBits e′″₁ = 11 e′″₂ = 12.5 e′″₃ = 13 e′″₄ = 14 . . . 00 1 3 1 2 . . . →3-bit left 01 1 2 4 4 . . . child index 10 2 2 1 2 . . . 11 1 3 5 5 . .. 00 1 1 4 4 . . . → 3-bit right 01 2 2 1 2 . . . child index 10 1 3 5 5. . . 11 3 2 1 2 . . .

Each of the 4^(th)-layer LUTs 881, 882 maps 2 information bits and a3-bit parent index to a 3-bit left child index and to a 3-bit rightindex. The 3-bit parent index identifies one of the eight possible8-dimensional pseudo-energies. The 2 information bits select from amongthe four ordered pairs that were averaged to yield the identified8-dimensional pseudo-energy. The 3-bit left child index is the index ofthe first 4-dimensional pseudo-energy in the ordered pair selected bythe 2 information bits, and the 3-bit right child index is the index ofthe second 4-dimensional pseudo-energy in the ordered pair selected bythe 2 information bits. For example, if the 3-bit parent indexidentifies the 8-dimensional pseudo-energy e′″3=13 and the 2 informationbits into the left and right child LUTs are equal to 10, the 3-bit leftchild index will identify the 4-dimensional pseudo-energy e″₁ and the3-bit right child index will identify the 4-dimensional pseudo-energye″₅. Each 3-bit child index is the 3-bit parent index of a respectiveone of the P-layer LUTs 841, 842, 843, 844.

5^(th) Layer LUT

There are 64 possible ordered pairs of the 8-dimensionalpseudo-energies. The pseudo-energy associated with a pair of8-dimensional pseudo-energies is the sum of the 8-dimensionalpseudo-energies. In this example, the 5^(th)-layer LUT 900 implements5-bit clipping. The 5-bit clipping is the act of discarding 32 of theordered pairs of the 8-dimensional pseudo-energies. The ordered pairsthat are discarded are those having the highest pseudo-energy associatedtherewith. The ordered pairs are discarded in the sense that the5^(th)-layer LUT 900 does not index the pseudo-energies associated withthose ordered pairs. This is referred to as “5-bit clipping” because 2⁵ordered pairs remain after the clipping operation.

In this example, the 5^(th)-layer LUT 900 implements 1-bit merging byconsidering the average of two ordered pairs of the 8-dimensionalpseudo-energies. For example, the “average pseudo-energy” resulting frommerging the two ordered pairs of 8-dimensional energies {(e′″₁; e′″₁),(e′″₁; e′″₂)} is 22.75. There are 16 such averages, which are indexed inthe 5^(th)-layer LUT 900. A pseudo-energy that is indexed in the5^(th)-layer LUT 900 is denoted a “16-dimensional pseudo-energy”. Thisis referred to as “1-bit merging”, because instead of using 5 bits toindex the not-discarded 32 ordered pairs of the 8-dimensionalpseudo-energies, only 4 bits are required to index the sixteen16-dimensional pseudo-energies.

An example of the 5^(th)-layer LUT 900 is given in Table 5:

TABLE 5 4 Info Bits Info Bit 1 16 16-Dim Pseudo-Energy e″″₁ = 22.75 . .. e″″₁₆  0 1 → 3-bit left 1 1 child index 0 1 → 3-bit right 1 2 childindex

The LUT 900 maps 5 information bits to a 3-bit left child index and to a3-bit right index. 4 of the information bits identify one of the 16possible 16-dimensional pseudo-energies. The other 2 information bitselects from among the two ordered pairs that were averaged to yield theidentified 16-dimensional pseudo-energy. The 3-bit left child index isthe index of the first 8-dimensional pseudo-energy in the ordered pairselected by the 1 information bit, and the 3-bit right child index isthe index of the second 8-dimensional pseudo-energy in the ordered pairselected by the 1 information bit. For example, if the 4 informationbits identify the 16-dimensional pseudo-energy e′″₁=22.75 and the 1information bit into the left and right child LUTs are both equal to 1,the 3-bit left child index will identify the 8-dimensional pseudo-energye′″₁ and the 3-bit right child index will identify the 8-dimensionalpseudo-energy e′″₂. Each 3-bit child index is the 3-bit parent index ofa respective one of the 4^(th)-layer LUTs 881, 882.

The preceding examples have equal bits into both child LUTs. This isonly one of the possible inputs. The information bits into the left andright child LUTs are typically different. All LUTs in any given layer ofthe tree will have the same number of client bits.

The scope of the claims should not be limited by the details set forthin the examples, but should be given the broadest interpretationconsistent with the description as a whole.

Appendix

Denote the alphabet of a QPSK constellation as S₁, S₂, S₃, S₄.

In each of the following quadruplets, the fourth QPSK symbol is uniquelydetermined by the values of the first three QPSK symbols. Let the threeQPSK symbols be denoted X₁, Y₁, and X₂, respectively. The fourth QPSKsymbol denoted Y₂ satisfies a condition X₁Y₁*+X₂Y₂*=0, where asterisk(*) denotes complex conjugate.

The first collection consists of the following 16 quadruplets of orderedQPSK symbols:

$\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix}$

The second collection consists of the following 16 quadruplets ofordered QPSK symbols:

$\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix}$

The third collection consists of the following 16 quadruplets of orderedQPSK symbols:

$\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix}$ $\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix}$

The fourth collection consists of the following 16 quadruplets ofordered QPSK symbols:

$\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{1}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{2}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix}$ $\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{4}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{4}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{3}} \\{Y_{1} = S_{1}} \\{X_{2} = S_{1}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{2}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{2}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix},\begin{pmatrix}{X_{1} = S_{4}} \\{Y_{1} = S_{3}} \\{X_{2} = S_{3}} \\Y_{2}\end{pmatrix}$

The Euclidean distance for any pair of quadruplets in a given collectionis greater than or equal to 2√{square root over (2)}.

1. An optical transmitter device comprising: a digital signal processor(DSP) comprising digital hardware, the DSP operative to generate shapedbits from a first set of information bits; to apply a systematic forwarderror correction (FEC) scheme to encode the shaped bits and a second setof information bits, where the first set of information bits and thesecond set of information bits are disjoint sets; and for multiple pairsof groups, each pair consisting of a first group and a second group,each first group consisting of a first unshaped bit and a first shapedbit and each second group consisting of a second unshaped bit and secondshaped bit to map the first unshaped bit and the first shaped bit to afirst quadrature phase shift keying (QPSK) symbol according to alabeling scheme; to map the second unshaped bit and the second one ofthe shaped bits to a second QPSK symbol according to the labelingscheme; and to map the first QPSK symbol and the second QPSK symbol tofour physical dimensions, wherein each unshaped bit is either one of thesecond set of information bits or one of multiple parity bits resultingfrom the FEC encoding; and wherein either the least significant bit inall groups is always one of the shaped bits or the most significant bitin all groups is always one of the shaped bits.
 2. The opticaltransmitter device recited in claim 1, further comprising a firstmodulator operative to modulate a first component of an optical carrierto convey the first QPSK symbol, the first component polarized in afirst polarization; and a second modulator operative to modulate asecond component of the optical carrier to convey the second QPSKsymbol, the second component polarized in a second polarization that isorthogonal to the first polarization.
 3. The optical transmitter devicerecited in claim 1, wherein the DSP is operative to apply any arbitraryshaping algorithm to the first set of information bits to generate theshaped bits.
 4. The optical transmitter device recited in claim 1,wherein the DSP is operative to apply a tree encoding structure to thefirst set of information bits to generate the shaped bits, the treeencoding structure having look-up tables that were programmed accordingto different pseudo-energies that were assigned to points of a two-pointconstellation.
 5. An optical transmitter device comprising: a digitalsignal processor (DSP) comprising digital hardware, the DSP operative togenerate shaped bits from a first set of information bits; to apply asystematic forward error correction (FEC) scheme to encode the shapedbits and a second set of information bits, where the first set ofinformation bits and the second set of information bits are disjointsets; and for multiple groups, each group consisting of K unshaped bitsand one shaped bit, to select one of two constellations based on a valueof the shaped bit, where all symbols of one of the two constellationshave an amplitude A, all symbols of the other of the two constellationshave an amplitude B, A and B are real positive numbers, B<A, and a sumof squares of the amplitudes A²+B² is equal to 1 or differs slightlyfrom 1, to use the K unshaped bits to select a first symbol from a firstof the two constellations and a second symbol from a second of the twoconstellations; where the first of the two constellations is selected,to map the first symbol to two physical dimensions of an optical carriercomponent that is polarized in a first polarization and to map thesecond symbol to two physical dimensions of an optical carrier componentthat is polarized in a second polarization that is orthogonal to thefirst polarization; and where the second of the two constellations isselected, to map the second symbol to the two physical dimensions of theoptical carrier component that is polarized in the first polarizationand to map the first symbol to the two physical dimensions of theoptical carrier component that is polarized in the second polarization,wherein each of the K unshaped bits is either one of the second set ofinformation bits or one of multiple parity bits resulting from the FECencoding, and K is a positive integer greater than three.
 6. The opticaltransmitter device recited in claim 5, where K equals 5, theconstellation whose symbols have the amplitude A is an 8 phase shiftkeying (8PSK) constellation, and the constellation whose symbols havethe amplitude B is a QPSK constellation.
 7. The optical transmitterdevice recited in claim 5, where K equals 4, and the two constellationsare QPSK constellations.
 8. The optical transmitter device recited inclaim 5, wherein the DSP is operative to apply any arbitrary shapingalgorithm to the first set of information bits to generate the shapedbits.
 9. The optical transmitter device recited in claim 5, wherein theDSP is operative to apply a tree encoding structure to the first set ofinformation bits to generate the shaped bits, the tree encodingstructure having look-up tables that were programmed according todifferent pseudo-energies that were assigned to points of a two-pointconstellation.
 10. An optical transmitter device comprising: a digitalsignal processor (DSP) comprising digital hardware, the DSP operative togenerate shaped bits from a first set of information bits; to apply asystematic forward error correction (FEC) scheme to encode the shapedbits and a second set of information bits, where the first set ofinformation bits and the second set of information bits are disjointsets; and for multiple groups, each group consisting of seven unshapedbits and one of the shaped bits to select one of two collections basedon a value of the one of the shaped bits; to use the seven unshaped bitsto select an ordered quadruplet of QPSK symbols from the selectedcollection; and to map the four QPSK symbols of the selected orderedquadruplet to eight physical dimensions, wherein each of the sevenunshaped bits is either one of the second set of information bits or oneof multiple parity bits resulting from the FEC encoding, and wherein alabeling scheme uses eight bits to label all 256 possible orderedquadruplets of QPSK symbols, one of the two collections consists of all128 ordered quadruplets of QPSK symbols whose 8-bit label according tothe labeling scheme has even parity, and the other of the twocollections consists of all 128 ordered quadruplets of QPSK symbolswhose 8-bit label according to the labeling scheme has odd parity. 11.The optical transmitter device recited in claim 10, further comprising afirst modulator operative to modulate, in a first time slot, a firstcomponent of an optical carrier according to a first of the four QPSKsymbols or a processed version thereof, the first component polarized ina first polarization, and to modulate, in a second time slot, the firstcomponent of the optical carrier according to a third of the four QPSKsymbols or a processed version thereof; and a second modulator operativeto modulate, in the first time slot, a second component of the opticalcarrier according to a second of the four QPSK symbols or a processedversion thereof, the second component polarized in a second polarizationthat is orthogonal to the first polarization, and to modulate, in thesecond time slot, the second component of the optical carrier accordingto a fourth of the four QPSK symbols or a processed version thereof. 12.The optical transmitter device recited in claim 10, wherein the DSP isoperative to apply any arbitrary shaping algorithm to the first set ofinformation bits to generate the shaped bits.
 13. The opticaltransmitter device recited in claim 10, wherein the DSP is operative toapply a tree encoding structure to the first set of information bits togenerate the shaped bits, the tree encoding structure having look-uptables that were programmed according to different pseudo-energies thatwere assigned to points of a two-point constellation.
 14. An opticaltransmitter device comprising: a digital signal processor (DSP)comprising digital hardware, the DSP operative to generate shaped bitsfrom a first set of information bits; to apply a systematic forwarderror correction (FEC) scheme to encode the shaped bits and a second setof information bits, where the first set of information bits and thesecond set of information bits are disjoint sets; and for multiplegroups, each group consisting of four unshaped bits and two of theshaped bits to select one of four collections based on a value of thetwo of the shaped bits; to use the four unshaped bits to select anordered triplet of QPSK symbols from the selected collection; and to mapfour QPSK symbols, including the QPSK symbols of the selected orderedtriplet, to eight physical dimensions, wherein each of the four unshapedbits is either one of the second set of information bits or one ofmultiple parity bits resulting from the FEC encoding, wherein athird-order Cartesian product of a QPSK constellation consists of all 64possible ordered triplets of QPSK symbols, and the four collections aredisjoint subsets of an augmented version of the third-order Cartesianproduct, each of the four collections consisting of a respective 16 ofthe ordered triplets, and wherein the QPSK symbols in the selectedordered triplet are denoted X₁, Y₁, and X₂, respectively, and anadditional QPSK symbol denoted Y₂ of the four QPSK symbols satisfies acondition X₁Y₁*+X₂Y₂*=0, where asterisk (*) denotes complex conjugate,such that a Euclidean distance between any pair of ordered quadrupletsin the collection is greater than or equal to 2√{square root over (2)}.15. The optical transmitter device recited in claim 14, furthercomprising a first modulator operative to modulate, in a first timeslot, a first component of an optical carrier according to a first ofthe four QPSK symbols or a processed version thereof, the firstcomponent polarized in a first polarization, and to modulate, in asecond time slot, the first component of the optical carrier accordingto a third of the four QPSK symbols or a processed version thereof; anda second modulator operative to modulate, in the first time slot, asecond component of the optical carrier according to a second of thefour QPSK symbols or a processed version thereof, the second componentpolarized in a second polarization that is orthogonal to the firstpolarization, and to modulate, in the second time slot, the secondcomponent of the optical carrier according to a fourth of the four QPSKsymbols or a processed version thereof.
 16. The optical transmitterdevice recited in claim 14, wherein the DSP is operative to apply anyarbitrary shaping algorithm to the first set of information bits togenerate the shaped bits.
 17. The optical transmitter device recited inclaim 14, wherein the DSP is operative to apply a tree encodingstructure to the first set of information bits to generate the shapedbits, the tree encoding structure having look-up tables that wereprogrammed according to different pseudo-energies that were assigned topoints of a two-point constellation.
 18. An optical transmitter devicecomprising: a digital signal processor (DSP) comprising digitalhardware, the DSP operative to generate shaped bits from a first set ofinformation bits; to apply a systematic forward error correction (FEC)scheme to encode the shaped bits and a second set of information bits,where the first set of information bits and the second set ofinformation bits are disjoint sets; and for multiple groups, each groupconsisting of six unshaped bits and one of the shaped bits to select oneof two collections based on a value of the one of the shaped bits; touse the four unshaped bits to select an ordered triplet of QPSK symbolsfrom the selected collection; and to map four QPSK symbols, includingthe QPSK symbols of the selected ordered triplet, to eight physicaldimensions, wherein each of the four unshaped bits is either one of thesecond set of information bits or one of multiple parity bits resultingfrom the FEC encoding, wherein a right-rotated QPSK symbol is aconventional QPSK symbol multiplied by exp(+iπ/8) and a left-rotatedQPSK symbol is a conventional QPSK symbol multiplied by exp(−iπ/8),wherein one of the two collections consists of all 64 possible orderedtriplets of right-rotated QPSK symbols, and the other of the collectionsconsists of all 64 possible ordered triplets of left-rotated QPSKsymbols, and wherein the QPSK symbols in the selected ordered tripletare denoted X₁, Y₁, and X₂, respectively, and an additional QPSK symboldenoted Y₂ of the four QPSK symbols satisfies a condition X₁Y₁*+X₂Y₂*=0,where asterisk (*) denotes complex conjugate.
 19. The opticaltransmitter device recited in claim 18, further comprising a firstmodulator operative to modulate, in a first time slot, a first componentof an optical carrier according to a first of the four QPSK symbols or aprocessed version thereof, the first component polarized in a firstpolarization, and to modulate, in a second time slot, the firstcomponent of the optical carrier according to a third of the four QPSKsymbols or a processed version thereof; and a second modulator operativeto modulate, in the first time slot, a second component of the opticalcarrier according to a second of the four QPSK symbols or a processedversion thereof, the second component polarized in a second polarizationthat is orthogonal to the first polarization, and to modulate, in thesecond time slot, the second component of the optical carrier accordingto a fourth of the four QPSK symbols or a processed version thereof. 20.The optical transmitter device recited in claim 18, wherein the DSP isoperative to apply any arbitrary shaping algorithm to the first set ofinformation bits to generate the shaped bits.
 21. The opticaltransmitter device recited in claim 18, wherein the DSP is operative toapply a tree encoding structure to the first set of information bits togenerate the shaped bits, the tree encoding structure having look-uptables that were programmed according to different pseudo-energies thatwere assigned to points of a two-point constellation.